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The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides

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Abstract

The problem of constructing the asymptotics of the Green function for the Helmholtz operator h 2Δ + n 2(x), x ∈ R n, with a small positive parameter h and smooth n 2(x) has been studied by many authors; see, e.g., [1, 2, 4]. In the case of variable coefficients, the asymptotics was constructed by matching the asymptotics of the Green function for the equation with frozen coefficients and a WKB-type asymptotics or, in a more general situation, the Maslov canonical operator. The paper presents a different method for evaluating the Green function, which does not suppose the knowledge of the exact Green function for the operator with frozen variables. This approach applies to a larger class of operators, even when the right-hand side is a smooth localized function rather than a δ-function. In particular, the method works for the linearized water wave equations.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526, Russia

    A. Yu. Anikin, S. Yu. Dobrokhotov & V. E. Nazaikinskii

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow oblast, 141700, Russia

    V. E. Nazaikinskii

  3. Aix Marseille Univ, Universite de Toulon, CNRS, CPT, Marseille, France

    M. Rouleux

Authors
  1. A. Yu. Anikin
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  2. S. Yu. Dobrokhotov
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  3. V. E. Nazaikinskii
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  4. M. Rouleux
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Correspondence to A. Yu. Anikin.

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Original Russian Text © A.Yu. Anikin, S.Yu. Dobrokhotov, V.E. Nazaikinskii, M. Rouleux, 2017, published in Doklady Akademii Nauk, 2017, Vol. 475, No. 6, pp. 624–628.

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Anikin, A.Y., Dobrokhotov, S.Y., Nazaikinskii, V.E. et al. The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides. Dokl. Math. 96, 406–410 (2017). https://doi.org/10.1134/S1064562417040275

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  • DOI: https://doi.org/10.1134/S1064562417040275

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